Let $f: X\to Y$ be an unramified double cover of an elliptic curve $Y$. Then $X$ is an elliptic curve as well. Is there a characterization of which vector bundles $\mathcal{F}$ on $Y$ can arise as the pushforward $f_*\mathcal{E}$ of a vector bundle $\mathcal{E}$ on $X$? Can we for example get a trivial vector bundle $\mathcal{O}_Y^r$ like this?
Since vector bundles on elliptic curves are classified and well understood, and since the map is very nice, I was hoping that one can say something. But I do not really understand how it behaves under this pushforward.
This is just an answer to your second question. No, the trivial bundle can not be direct image of a vector bundle on $X$.
To see this, assume there is a vector bundle of rank $r$, $E$ on $X$ such that $f_*E$ is trivial of rank necessarily $2r$. Then $\chi(Y, f_*E)=0$ and thus $\chi(X,E)=0$. So degree of $E$ is zero by Riemann-Roch. But $f^*f_*E\to E$ is onto, and thus $E$ is globally generated. A globally generated degree zero vector bundle must be trivial and thus $E$ is trivial of rank $r$ and thus $H^0(E)=r\neq 2r=H^0(f_*E)$, a contradiction.
For your first question, one has $f_*\mathcal{O}_X=\mathcal{O}_Y\oplus L$ for a non-trivial line bundle $L$ which is 2-torsion. Thus, for any vector bundle $E$ on $X$, one has $L\otimes f_*E\cong f_*E$. I think this is the only condition you need, though I haven't checked it carefully. So, the vector bundles on $Y$ which are direct images of vector bundles on $X$ should be those $G$ on $Y$ with $L\otimes G\cong G$.